We concentrated on the development of an algebraic framework
allowing for a mathematical represention
of networks regardless of their dimension and the underlying stochastic
assumptions.
This formalism also allows one to define classes of networks
(e.g. event graphs, free choice nets etc.)
for which similar methods of analysis work.
Based on this mathematical representation,
one can then design controls or evaluate analytically
performaces in new ways, quite different from
the more classical Markovian approach.
For more on this approach, see the
Wiley book entitled
Synchronization and Linearity, which
we coauthored with
G. Cohen,
G. J. Olsder
and
J. P. Quadrat
in 1992.

Concerning my own research, here are the main domains of current interest,
which are also summarized in the
slides of a lecture that I gave at the
10-th Applied
Probability Conference in Ulm (Germany) in July 1999:

STOCHASTIC ANALYSIS: an important issue in this setting is that
of the computation and analytical characterization of the
stationary distributions in networks. Here are the main directions:

ANALYTIC EXPANSIONS for (max,plus) Lyapunov exponents with
D. Hong:
see the reports
RR-3427.
RR-3558.
The aim here is to work out expansions for the
throughput of closed networks with random services.

COMPUTATION OF DISTRIBUTIONS in (max,plus) linear systems: mean values in
RR-2494,
with
V. Schmidt
, Laplace transforms and transient behavior in
RR-2785 and
RR-3022,
with V. Schmidt and S. Hasenfuss from
Ulm University.
This concerns open networks with Poisson input point processes,
for which we wish to derive expressions for the law of
stationary or transient waiting times.

HEAVY TAIL ASYMPTOTICS for the stationary distributions in
various classes of networks:

for irreducible (max,plus)
networks, see the Questa 99 paper
in collaboration with
V. Schmidt and S. Schlegel;

for monotone separable networks, see
RR-4197
in collaboration with
S. Foss; in this paper (now published in the Annals of Applied
Probability), we started developing the theory of rare events
for networks and we worked out some examples, in particular
we derived the exact asymptotics for tandem queues with subexponential
service times.

The tail asymptotics of subexponential (max,plus) networks
was then derived in RR-4952
and that of generalized Jackson networks in
RR-5081,
jointly with
S. Foss and
M. Lelarge.

There are several quite interesting developments
which lead to a more and more complete theory of rare events for networks.
A comprehensive exposition of the advances on the matter
can be found in the thesis (defended in 2005) and the web page
of Marc Lelarge.

ALGEBRA: we worked on the algebraic framework for
non homogenous linear systems in
RR-3435
jointly with
R. Agrawal
of University of Wisconsin and R. Rajan of AT&T labs. One of the motivations
for this is to model integrated service networks.

NON-LINEAR NETWORKS: beyong the linear classes,
there are several other classes of
networks which can be analyzed via certain extensions
of this algebraic framework.
The main results bear on

The MONOTONE-SEPARABLE framework that we introduced in the
95 JAP paper
jointly with
S. Foss;
this class contains as special cases generalized Jackson networks,
(max,plus) linear networks, multiserver queues and many classes
of Free Choice stochasttic Petri Nets.

(MAX,PLUS) LINEAR SYSTEMS WITH FIFO PERTURBATIONS. The report
RR-3434
with
T. Bonald
.
bears on the analysis of the stability region for the TCP protocol.

GENERALIZED JACKSON NETWORKS, a topic that we started investigating
jointly with
S. Foss
(when he was still at
Novosibirsk University) using
ergodic theory RR 2015 ; see
in particular the following paper which
was published in the proceedings of the Edinburgh Stochastic
Network conference.

Jointly with
Serguei Foss we studied infinite dimensional
max plus systems based on random closed sets of the Euclidean plane in the paper
Poisson Hail on a Hot Ground,
to appear in Advances in Applied Probability in 2011.

SIMULATION: our main contribution bears on the parallel simulation
of event-graphs (see
RR-1520
with M. Canales). For the extension to
free-choice nets and other more general classes, see the
web pages of
N. Furmento,
B. Gaujal and
A. Jean-Marie.

Our research group was partner of the European project
Alapedes
which was part of the
TMR program;
there was also a Spring school on this topic
in Noirmoutier:
Algèbres Max-Plus et applications en informatique et automatique
which stressed the recent developments of the theory
and several domains of application in manufacturing, in
transportation systems, and in computer science.

Synchronization mechanisms in parallel
processing (see the
Qmips book, which we co-edited with
A.
Jean-Marie and
I. Mitrani,
which summarizes the
results that we obtained in a BRA European project on the matter).